Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$?
My feeling is "yes", but I'm not sure if we need the extension $K/k$ to be abelian. My intuition would be that the "infinite" part of $Gal(K/k)$ comes from at least one copy of $\mathbb{Z}_p$. Am I right about this?
Thank you in advance!
Update (11/21/2013): In response to hunter's answer, I want to ask, under which conditions does a $p$-extension contain the $\Bbb Z_p$-extension? Update (28/11/2013): The question above is possibly too broad and not answerable.
No, take the compositum of all quadratic extensions of $\mathbb{Q}$.Every element of the Galois group is $2$-torsion, so this cannot possibly have a copy of $\mathbb{Z}_2$ as a quotient. If you allow only finitely many primes to ramify, then the answer becomes yes (because your Galois group is then finitely topologically generated, and now appeal to the structure theory of $p$-groups).